| L(s) = 1 | + 3-s − 3.23·5-s − 0.236·7-s + 9-s + 1.23·11-s + 3.47·13-s − 3.23·15-s − 4·17-s − 0.236·21-s − 3.23·23-s + 5.47·25-s + 27-s − 1.52·29-s + 4.70·31-s + 1.23·33-s + 0.763·35-s + 7·37-s + 3.47·39-s − 2.47·41-s − 6.23·43-s − 3.23·45-s − 2·47-s − 6.94·49-s − 4·51-s − 12.1·53-s − 4.00·55-s + 3.70·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.44·5-s − 0.0892·7-s + 0.333·9-s + 0.372·11-s + 0.962·13-s − 0.835·15-s − 0.970·17-s − 0.0515·21-s − 0.674·23-s + 1.09·25-s + 0.192·27-s − 0.283·29-s + 0.845·31-s + 0.215·33-s + 0.129·35-s + 1.15·37-s + 0.555·39-s − 0.386·41-s − 0.950·43-s − 0.482·45-s − 0.291·47-s − 0.992·49-s − 0.560·51-s − 1.67·53-s − 0.539·55-s + 0.482·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 4.70T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 - 0.236T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 3.47T + 73T^{2} \) |
| 79 | \( 1 - 8.70T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 3.70T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66803691484786354237330938802, −6.59710349922343494274312298454, −6.46689067296174987919532991949, −5.16253712460357598094782771442, −4.35818300936969460260950290767, −3.83903873452485684301066282633, −3.28524480456188948746071594263, −2.29886533864765185381890586651, −1.20090534844219592675348220209, 0,
1.20090534844219592675348220209, 2.29886533864765185381890586651, 3.28524480456188948746071594263, 3.83903873452485684301066282633, 4.35818300936969460260950290767, 5.16253712460357598094782771442, 6.46689067296174987919532991949, 6.59710349922343494274312298454, 7.66803691484786354237330938802
